Theorem carden2b 9896

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9895 are meant to replace carden 10480 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 9894	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2		ennum 9876	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
3	2	biimpa 476	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4		cardid2 9882	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6		ensym 8951	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ≈ 𝐴)
8		entr 8954	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (card‘𝐵) ≈ 𝐴)
9	5, 7, 8	syl2anc 584	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
10	1, 9	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11		cardon 9873	. . . . 5 ⊢ (card‘𝐴) ∈ On
12		cardon 9873	. . . . 5 ⊢ (card‘𝐵) ∈ On
13		ontri1 6354	. . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
14	11, 12, 13	mp2an 692	. . . 4 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
15	10, 14	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16		cardne 9894	. . . . 5 ⊢ ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17		cardid2 9882	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18		id 22	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵)
19		entr 8954	. . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (card‘𝐴) ≈ 𝐵)
20	17, 18, 19	syl2anr 597	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
21	16, 20	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22		ontri1 6354	. . . . 5 ⊢ (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
23	12, 11, 22	mp2an 692	. . . 4 ⊢ ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
24	21, 23	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
25	15, 24	eqssd 3961	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26		ndmfv 6875	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
27	26	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
28	2	notbid 318	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
29	28	biimpa 476	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30		ndmfv 6875	. . . 4 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
32	27, 31	eqtr4d 2767	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
33	25, 32	pm2.61dan 812	1 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102  dom cdm 5631  Oncon0 6320  ‘cfv 6499   ≈ cen 8892  cardccrd 9864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-er 8648  df-en 8896  df-card 9868

This theorem is referenced by:  card1  9897  carddom2  9906  cardennn  9912  cardsucinf  9913  pm54.43lem  9929  nnadju  10127  nnadjuALT  10128  ficardun  10130  ackbij1lem5  10152  ackbij1lem8  10155  ackbij1lem9  10156  ackbij2lem2  10168  carden  10480  r1tskina  10711  cardfz  13911